3.Let X1, X2, … , Xnbe a random sample from the distribution with probability mass function P(Xi= 1)= θθ3+, P(Xi= 2)= θ32+, P(Xi= 3)= θ31+, θ> 0. a) Find a sufficient statistic for θ. b) Obtain the method of moments estimator θ~of θ. c) Obtain the maximum likelihood estimator θˆof θ. 4.A store sells "16-ounce" boxes of Captain Crispcereal. A random sample of 9 boxes was taken and weighed. The results were the following (in ounces): 15.5 16.2 16.1 15.8 15.6 16.0 15.8 15.9 16.2 Assume the weight of cereal in a box is normally distributed. Hint: Σx= 143.1, Σx2= 2275.79, Σ(x– x)2= 0.50. a) Compute the sample mean xand the sample standard deviation s. b) Construct a 95% confidence interval for the overall average weight of boxes of Captain Crispcereal. c) Construct a 95% confidence upper bound for the overall average weight of boxes of Captain Crispcereal. d) Construct a 90% confidence interval for the overall standard deviation of the weights of boxes of Captain Crispcereal. e) Construct a 99% confidence lower bound for the overall standard deviation of the weights of boxes of Captain Crispcereal. 5.Starting annual salaries for college graduates of certain majors are believed to have a standard deviation of approximately $1800. What is the minimum required sample size to estimate the average annual salary for college graduates to within $300 with 90% confidence?

6.A researcher wishes to determine whether the starting salaries of high-school math teachers in private schools are different from those of high-school math teachers in public schools. She selects a sample of new math teachers from each type of school and calculates the sample means and sample standard deviations of their salaries. Assume that the populations are normally distributed and the population variances are equal. Construct a 95% confidence interval for the difference in average starting salaries of high-school math teachers in private and public schools.

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